10.1098/rsta.2001.0808 Beyond simple depletion: phase behaviour of colloid–star polymer mixtures By W. C. K. Poon1, S. U. Egelhaaf1, J. Stellbrink1,2, J. Allgaier2, A. B. Schofield1 and P. N. Pusey1 1Department of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, UK 2IFF-Neutronenstreuung II, Forschungszentrum J¨ulich GmbH, D-52425 J¨ulich, Germany Significant progress has been made in the last decade in understanding mixtures of hard-sphere colloids and (smaller) non-adsorbing, ideal, linear polymers. We intro- duce extra complexity into this simple model system by replacing the linear polymers with star-branched polymers with increasing functionality but constant radius of gyration. The observed phase diagrams, interpreted in light of what is known about hard-sphere colloid plus linear polymer and binary-hard-sphere mixtures, suggest that 32-arm stars are close to behaving hard-sphere-like in colloid–star mixtures at this size ratio. Keywords: depletion; colloid; hard spheres; polymer; star polymer 1. Introduction Colloids, polymers and surfactants almost always occur in the form of mixtures, whether in nature or as industrial products. One of the main tasks of soft con- densed matter physics is to give generic (i.e. chemical-details-independent) insight into the structure, dynamics, phase behaviour and non-equilibrium properties of such mixtures. In practice, this entails identifying and studying in detail model systems stripped of as many extraneous features as possible. Phase transitions in ‘soft mixtures’ can be driven mainly by enthalpy or entropy. Demixing in mixtures of polymers is the outstanding example of enthalpy-driven phenomena. In other cases, entropy dominates. Over the last decade, a system that has emerged as a model for such entropy-driven phase transitions is a mixture of hard-sphere colloids and (smaller) random-coil polymers in a near-theta solvent for the latter. In this system, the only interaction is that of excluded volume between the colloids and between the colloids and polymers; the polymer coils can be treated, to a first approximation, as ideal, and therefore interpenetrate each other freely. Detailed studies by experiment, theory and simulation have led to substantial progress in understanding phase transitions and metastability in this idealized model. The essential physics is captured by the ‘depletion’ picture first proposed by Asakura & Oosawa (1958) and later independently by Vrij (1976). The centres of mass of poly- mer molecules are excluded from the vicinity of each particle, creating a ‘depletion zone’. Overlap of the depletion zones from neighbouring particles creates extra vol- ume for the polymers, thus increasing their entropy and lowering the free energy Phil. Trans. R. Soc. Lond. A (2001) 359, 897–907 c 2001 The Royal Society 897 898 W. C. K. Poon and others of the system. It turns out that the topology of the equilibrium phase diagram depends on the relative sizes of the polymer and colloid. In addition, a rich ‘zoo’ of non-equilibrium behaviour has been identified and rationalized within an emerging general framework. In this paper, we report experiments that go systematically beyond the ideal- ized model: the nearly ideal linear polymer is replaced by star-branched polymers of increasing functionality (number of arms). To simplify the terminology, we will refer to the model system and the more complex systems just introduced as the colloid–polymer mixture and the colloid–star mixture, respectively. Moreover, when ‘polymer’ is used without qualification, it refers to a linear coil. Our generic aim is to see how far the emerging comprehensive understanding of colloid–polymer mix- tures can aid the interpretation of phenomena in more complex systems. The specific motivation for choosing to study colloid–star mixtures is to observe the way star poly- mers of increasing functionality become progressively less like interpenetrable coils and more like mutually excluding hard particles (Seghrouchni et al. 1998). Thus, studying colloid–star mixtures with different functionalities can tell us how phase behaviour evolves between the extremes of a colloid–polymer mixture on the one hand, and a binary hard-sphere colloid on the other. Below, we first review the phase behaviour and non-equilibrium properties of colloid–polymer mixtures in § 2. In § 3 we report the phase diagrams of colloid– star mixtures with stars of functionalities 2, 6, 16 and 32 but the same radii of gyration (thus maintaining a constant star-to-colloid size ratio). The data are then discussed in § 4 in terms of existing knowledge of colloid–polymer and binary hard- sphere mixtures. We conclude in § 5 with some speculations on colloid–star mixtures with stars of higher functionality, and suggestions of other areas of exploration that go systematically beyond simple depletion. 2. Colloid–polymer mixtures: a brief review The theoretical prediction for the phase behaviour of a colloid–polymer mixture is by now well known (Gast et al. 1983; Lekkerkerker et al. 1992). The key parameter controlling the topology of the phase diagram is the ratio of the size of the polymer, e.g. as measured by its radius of gyration (rg), to the radius of the colloid (R), ξ = rg/R. When ξ is less than a certain critical value, ξc, the addition of polymer merely expands the fluid–crystal coexistence region of pure hard spheres, which occurs at 0.494 <φc < 0.545 (where φc is the colloid volume fraction). At ξ>ξc, a colloidal liquid phase becomes possible, and the phase diagram displays a colloidal gas–liquid critical point and a region of three-phase coexistence of colloidal gas, liquid and crystal. Mean-field theories predict ξc ≈ 0.33. Computer simulations confirmed this picture (Meijer & Frenkel 1994; Dijkstra et al. 1999). The phase diagrams for ξ =0.1 and 0.5 calculated according to Lekkerkerker et al. (1992) are shown in figure 1. Experimentally, the most well-studied realization of this simple model system is a mixture of sterically stabilized PMMA spheres and linear polystyrene (PS) dispersed in cis-decahydronaphthalene (cis-decalin) (Ilett et al. 1995). These PMMA particles behave as nearly perfect hard spheres (Underwood et al. 1994), while cis-decalin is a theta-solvent for PS at 13 ◦C (Berry 1966). The qualitative picture outlined above was confirmed, although there were quantitative differences between experiment and theory. Experimentally, ξc ≈ 0.25, and theory predicts concentrations of polymer in Phil. Trans. R. Soc. Lond. A (2001) Beyond simple depletion 899 0.5 (a) 0.4 0.3 η 0.2 F + C 0.1 F 0 0.2 0.4 0.6 0.8 φ 1.0 (b) 0.8 G + C 0.6 G + L + C η 0.4 G + L L + C 0.2 F 0 0.1 0.2 0.3 0.4 0.5 φ Figure 1. Theoretical phase diagrams of a mixture of non-adsorbing, ideal linear polymers (radius of gyration rg) with hard spheres (radius R). The phase diagram topology depends on the size ratio ξ = rg/R:(a) ξ =0.1 and (b) ξ =0.5. The horizontal and vertical axes are the colloid volume fraction, φ, and the polymer volume fraction, η (calculated using rg), respectively. G denotes gas, L denotes liquid, F denotes fluid and C denotes crystal. the dense colloidal phases that are too high by up to two orders of magnitude. The structure of the liquid phase in systems with ξ ξc has been measured (Moussa¨ıd et al. 1999) and calculated (Louis et al. 1999; Dijkstra et al. 1999; Fuchs & Schweizer 2000). The non-equilibrium behaviour of colloid–polymer mixtures has also been studied in detail. For all values of ξ studied, samples with the highest concentrations of polymer failed to phase separate according to the predictions of equilibrium theory, but produced ‘transient gels’: space-filling ramified networks of particles that collapse after finite time to give denser, amorphous sediments. The most well-studied transient gels (Poon et al. 1995; Evans & Poon 1997; Evans et al. 1997) are those formed in mixtures with small polymers, ξ ∼ 0.1 or less. The formation and initial structure of Phil. Trans. R. Soc. Lond. A (2001) 900 W. C. K. Poon and others such gels have been discussed in terms of diffusion-limited cluster aggregation with finite bond energies (Haw et al. 1994, 1995a, b, 1997). The position of the gel line can be predicted with reasonable accuracy within the frameworkof mode-coupling theories of ergodicity breaking (Bergenholtz & Fuchs 1999). However, the long-time evolution and eventual collapse of these transient gels is still not fully understood (Poon et al. 1999a). Finally, the kinetic pathways followed by systems with ξ>ξc phase separating into coexisting colloidal gas, liquid and crystal phases have also been identified and rationalized in terms of the ‘free energy landscape’ of the system (Poon et al. 1999b, 2000). The literature of colloid–polymer mixtures is often cited in discussions of binary hard spheres with small ξ (say, 0.2 or less), now defined as the ratio of the radii of the two species of hard spheres ξ = R1/R2 (with R1 <R2). (We will use ξ to denote all size ratios in this paper; the meaning in each case should be clear from the context.) Published experimental phase diagrams at ξ ∼ 0.1 disagree (see, for example, van Duijneveldt et al. 1993; Imhof & Dhont 1995; Dinsmore et al. 1995). Recent theoretical work(Louis et al. 2000a) suggests that this may reflect the extreme sensitivity of the phase behaviour to the form of the interaction between the two species, and therefore to the exact nature of the colloids used experimentally to model hard spheres. Small deviations from perfect hardness in the mutual interaction can be modelled as ‘non-additivity’.